Rowling may have claimed "close to the speed of light," but I got the impression even from the books that the objects visibly fly through the air instead of, by the human eye, appearing to teleport into your hands. "‘Accio! Accio! Accio!’ she shouted, and toffees zoomed from all sorts of unlikely places, including the lining of George’s jacket..."
– KaiMar 18 at 15:59

2 Answers
2

They fly slowly enough to be easily caught.

Though objects summoned by Accio travel fairly quickly, they travel slowly enough that they can be easily caught. Harry Summons a dictionary and is able to catch it without injury, despite the dictionary that flies towards him being fairly heavy.

“Accio Dictionary!’

The heavy book soared out of Hermione’s hand, flew across the room, and Harry caught it.

‘Harry, I really think you’ve got it!’ said Hermione, delightedly.” - Harry Potter and the Goblet of Fire, Chapter 20 (The First Task)

Therefore, objects summoned by Accio can’t possibly be traveling too fast, because it’s possible to catch them without being injured by the speeding object.

They don’t always arrive instantly.

Objects summoned by Accio don’t always get to the wizard who summoned them instantly. When Harry Summons his Firebolt, he has to wait a bit for it to actually get to him, making him wonder if he’d Summoned it successfully until he heard it.

“Accio Firebolt!’ he shouted.

He waited, every fibre of him hoping, praying … if it hadn’t worked … if it wasn’t coming … he seemed to be looking at everything around him through some sort of shimmering, transparent barrier, like a heat haze, which made the enclosure and the hundreds of faces around him swim strangely …

And then he heard it, speeding through the air behind him; he turned and saw his Firebolt hurtling towards him around the edge of the woods, soaring into the enclosure, and stopping dead in mid-air beside him, waiting for him to mount.” - Harry Potter and the Goblet of Fire, Chapter 20 (The First Task)

Therefore, Accio doesn’t Summon objects in a way that gets them to the caster near-instantly. Additionally, this implies that the speed of an object summoned by Accio doesn’t increase for an object that has a longer distance to travel to get it to its destination more quickly and thus get it to the caster just as quickly as a closer object. That means the further the object is from the wizard Summoning it, the longer it’ll take it to arrive using Accio.

They don’t travel close to light speed.

Accio can’t possibly Summon objects at close to the speed of light, despite JKR’s statement that they do. For it to be true, objects Summoned by Accio would be traveling at a lethally dangerous speed m as well as invisible to the human eye, which clearly isn’t the case.

How "Fast" is the Speed of Light?

Light travels at a constant, finite speed of 186,000 mi/sec. A traveler, moving at the speed of light, would circum-navigate the equator approximately 7.5 times in one second. By comparison, a traveler in a jet aircraft, moving at a ground speed of 500 mph, would cross the continental U.S. once in 4 hours. - How "Fast" is the Speed of Light? (NASA)

The speed of light is many times faster than the speed of a bullet, so objects traveling close to that speed would be highly dangerous.

Accio makes objects travel at "close to the speed of light"

Why couldn’t Newt use ‘Accio’ to retrieve all his beasts?
‘Accio’ only works on inanimate objects. While people or creatures may be indirectly moved by ‘Accio-ing’ objects that they are wearing or holding, this carries all kinds of risks because of the likelihood of injury to the person or beast attached to an object travelling at close to the speed of light.JK Rowling's New Website - FAQ